[sidkshatriya]

I think Falcon heavy is not needed. Webb's weight is ~6200 kg. Falcon 9 is 8300 kg to GTO (Geostationary transfer orbit). This leaves about ~2000 kg fuel equivalent to spare which should be more than enough for the special location Webb is aiming for.

But yes, the contract for the webb launch was probably locked a very long time ago.

(Though it's also possible that Falcon 9's fairing wont be able to accommodate Webb)