Two-copy distillability of the two-ququart Werner state at $\alpha = -1/2$

Show that the $d = 4$ Werner state $\rho(4, -1/2)$ is not 2-copy distillable: for all rank-2 projectors P, Q acting on two copies of each party's space, $\left(P\,\operatorname{tensor}\,Q\right)\left[(\rho^\Gamma)^{\operatorname{tensor} 2}\right]\left(P\,\operatorname{tensor}\,Q\right)$ has no negative eigenvalue; or exhibit P, Q giving a negative eigenvalue. Its partial transpose is proportional to a unitary (Problem 5 of Horodecki, Rudnicki and Zyczkowski 2022).

A proof of positivity over all rank-2 projector pairs, or explicit projectors giving a negative eigenvalue.