It is known that p-blocks of symmetric groups can be characterized by the set of partitions with the same p-core. The following problem comes from the representation of symmetric groups, which is related to the intersection of two blocks of a given symmetric group:

Let $p>q$ be two prime numbers. Let $\lambda$ be a partition whose $p$-core is $\lambda_p$ and $q$-core is $\lambda_q$. Assume that $|\lambda|>|\lambda_p|+|\lambda_q|$. Is it true that there always exists some partition $\mu\neq \lambda$ with $|\mu|=|\lambda|$, such that $\mu$ also has $p$-core $\lambda_p$ and $q$-core $\lambda_q$?

I have verified this conjecture for partitions with small sizes. But I can't prove it in general. Any hint will be appreciated.

Let $p>q$ be two prime numbers. Let $\lambda$ be a partition whose $p$-core is $\lambda_p$ and $q$-core is $\lambda_q$. Assume that $|\lambda|>|\lambda_p|+|\lambda_q|$. Is it true that there always exists some partition $\mu\neq \lambda$ with $|\mu|=|\lambda|$, such that $\mu$ also has $p$-core $\lambda_p$ and $q$-core $\lambda_q$?

I have verified this conjecture for partitions with small sizes. But I can't prove it in general. Any hint will be appreciated.