Linear upper bound for the partition rank of two maximally entangled pairs

Establish whether the partition rank of two copies of a maximally entangled state, with allowed partitions $\{AC|BD,AD|BC\}$, admits an upper bound that scales linearly with the local dimension $d$.

Background

The paper considers the tensor product of two maximally entangled states, $|\Phi^+\rangle_{AB}|\Phi^+\rangle_{CD}$, and asks for the minimum number of terms in a decomposition whose terms are separable only across the partitions $AC|BD$ or $AD|BC$.

A construction gives the upper bound $1+d(d-1)/2$, which is quadratic in the local dimension. The authors identify as unresolved whether a stronger upper bound with linear dependence on $d$, as suggested in the cited mathematical literature, can be obtained.