Maximal even $(\bar p,\bar q)$-core partition conjecture

Let $p$ and $q$ be distinct odd primes. A partition is a $(\bar p,\bar q)$-core partition if it is simultaneously a $\bar p$-core and a $\bar q$-core, and it is even when its corresponding parity is even. Maximal even core conjecture. There exists a $(\bar p,\bar q)$-core partition $\lambda$ such that every even $(\bar p,\bar q)$-core partition is contained in $\lambda$.

This asks whether the even core partitions admit a common largest member under containment, analogous to the known result for all $(\bar p,\bar q)$-core partitions. The source presents this as a future direction and gives no resolution.