The Generalized Distillability Conjecture for critical Werner states

Let $m\geq1$, let $\mathcal{H}_k=\mathcal{H}_k^A\otimes\mathcal{H}_k^B$ have equal finite local dimensions $d_k$, and let $F_k$ be the flip operator on $\mathcal{H}_k$. Define the critical Werner state $\rho_k^W=1-F_k/2$ and the generalized critical Werner state

$$\rho_{(d_1,\ldots,d_m)}^W=\rho_1^W\otimes\rho_2^W\otimes\cdots\otimes\rho_m^W.$$

Generalized Distillability Conjecture. All generalized critical Werner states $\rho_{(d_1,\ldots,d_m)}^W$, $m\geq1$, are $1$-indistillable.

The case $m=1$ is known because critical Werner states are $1$-indistillable, but the assertion for arbitrary numbers and dimensions of tensor factors is open. Since $m$ and the $d_k$ are arbitrary, the conjecture also implies the Distillability Conjecture.