> Until now there was no such correlation with Alzeimher's and Fungi.
The point of the criticism seems to be that with an improper control group, there still isn't such a correlation, just a suggestion of a possibility of a correlation. But it might be a correlation between, e.g., age and fungal infection.
That's why, but for the variables under study, you want your control group to look as much like your experimental group as possible.
A general correlation between age and fungal infection might be even more significant, since that suggests a path to reduce the effects of aging in general. Either way, a follow-up study to remove those control flaws and hone in on the cause would be necessary.
Even with an improper control group, the fungi-Alzheimer's correlation still exists. There may be another factor which is causing the correlation (a so-called 'dreaded third thing'), and the correlation may not indicate causation, but the correlation still exists.
The point of the criticism seems to be that with an improper control group, there still isn't such a correlation, just a suggestion of a possibility of a correlation. But it might be a correlation between, e.g., age and fungal infection.
That's why, but for the variables under study, you want your control group to look as much like your experimental group as possible.