The ice in Antarctica and Greenland is not going to melt altogether and it would take thousands of years for it to melt away at current pace even if trends aren't reversed by another series of similar volcanic events.
Not at ice divides (like hydrological divides) where ice cores are taken (ice cores are almost always taken at divides because the ice flows apart, so the stratigraphy is not as disturbed). It's too cold to melt out there anytime soon. These are high-elevation locations, so there is either no melting (East Antartica ones ~3000 m) or extremely minimal melting, rarely, during the height of summer (Greenland and some other places in Antarctica). In Antarctica, surface melt is a rounding error everywhere, and response to climate change is almost entirely due to ice sheet-ocean interactions with a warming ocean and changing currents, and a thing called the marine ice-sheet instability.
At any rate, these locations will always be accumulation zones where more snow falls then melts (with the mass balance preserved by diverging flow so as to continuously thin the ice, to approximately compensate), with melt events occuring only rarely. The concern there is that meltwater could contaminate near-surface compacted snow (firn) layers, but even then it wouldn't be enough to do much. As such, stratigraphy will be reasonably preserved even if the ice sheet as a whole draws down significantly (including at divides).
Really extreme climate change (which is in the cards long-term) could eventually change this, especially if the ice sheets reduce enough in height such that a significant ice sheet-elevation feetback occurs. However, we'd have much bigger things to worry about at that point.
You wrote 'feetback' instead of 'feedback', and Colemak is the only popular layout (vs Qwerty and Dvorak) where 't' and 'd' are right next to each other.
We have the most extensive data available for recent years, though. Ice core samples are also nothing new, they've been extensively studied for decades. Many of these cores are carefully removed and placed in ice cold storage warehouses: https://qz.com/1590747/an-antarctic-ice-core-may-show-1-5-mi...
Again, the physical ice core samples are permanently kept in giant freezer warehouses. And we know how to store binary archive data for thousands of years securely.
In 10000 years the earths crust moves warehouses are destroyed. The discovered binary has to be decoded and translated. Humankind has changed, evolved or devolved in ways we cannot yet know. Looking forward from now the two proposed future reality are equally valid. We have no way of knowing that outcome.
The ice in Greenland is up to 3000 meters thick, and Antarctica is up to almost 5000 meters. Are you suggesting the net melt could average up to half a meter a year?
"Thousands of years" is clearly wrong, so it is at best several centuries.
About "decades": I'm not saying it will, but that it could.
It is completely in the plausible, and scientists have been proven to be too much conservative, to the great surprise of everyone including themselves.
That says that the fastest plausible scenario has the ice sheet melting in 1000 years, while the greater likelihood is 10,000 years. And that's if/after we cross temperature thresholds we're not guaranteed to cross.
Nowhere does it entertain the whole thing going in decades. If you have credible sources that argue for the possibility of decades, or even a century or two, you should add that to the page.
The tipping point for Greenland is said to be 2.5°C, and at that tipping point the minimum melting time is said to be 1000 years.
Apart from the fact that all of this relies only on the data we have (we'll discover many more chaotic systems and tipping points, amplifying feedback etc), it also says nothing about what happens at 3.5°C or 4°C warming for Greenland (important precision, as Greenland itself is warming at a much larger rate than the planet globally).
And no, I have no reference to add, only to be cautious and reminding that everybody has been way too much optimistic.
And I want to precise that I am not speaking about Antarctica.
I don't think that Antarctica will lose its ice cover. Only probably its ice shelves, and also some glaciers will shrink, but overall it'll stay white.