Conjecture that p-SKT and p-Kähler manifolds coincide under the ∂∂̄-lemma

Let $X$ be a compact complex manifold of complex dimension $n$ satisfying the ∂∂̄-lemma, and let $p$ be an integer with $1\leq p\leq n-1$. The properties $p$-SKT and $p$-Kähler are the degree-$p$ strong Kähler-with-torsion and Kähler-type positivity properties, respectively.

$p$-SKT/$p$-Kähler conjecture.

$$X\text{ is }p\text{-SKT}\quad\Longleftrightarrow\quad X\text{ is }p\text{-Kähler},\qquad 1\leq p\leq n-1.$$

The conjecture seeks an equivalence between these two degree-$p$ structures under the ∂∂̄-lemma. The source notes that $p$-SKT is stable under small deformations in this setting, but does not state that the conjectured equivalence is known.