The Morita-class count conjecture for weight-two blocks of symmetric groups

Let $p$ be a prime, and let $M(p,w)$ denote the number of Morita equivalence classes of $p$-blocks of symmetric groups with weight $w$. The blocks of weight $2$ are being considered; computations confirm the proposed formula for $p\leq 11$.

Weight-two Morita-class count conjecture.

$$M(p,2)=\frac{1}{2p}\binom{2p}{p-1}+\frac{1}{2}\binom{p}{\lfloor p/2\rfloor}.$$

This conjecture asserts sharpness of the known upper bound in weight $2$. It has been confirmed computationally for primes $p\leq 11$, but the general case remains open.