Two conjectures on highly composite numbers. Conjectures no. 34 and 35

Conjecture no. 34

There is finitely many highly composite numbers such that if one is added to ,it it is a perfect square.

Conjecture no. 35

Is the square root of the highly composite numbers that satisfy conjecture no. 34 always prime if not is it prime infinitely often.

https://en.m.wikipedia.org/wiki/Highly_composite_number

https://oeis.org/A002182

Thank you,

Nunghead