The square bracket polarized partition characterization of the selection property S₁(Ω, Ω)

Let $(X,\tau)$ be a topological space. Write

$$\left(\begin{array}{c} \Omega\\ \Omega \end{array}\right) \longrightarrow \left[ \begin{array}{c}\Omega\\ \Omega \end{array}\right]^{1,1}_{k/<3}$$

when the square bracket polarized partition relation holds for the families of $\Omega$-covers of $X$, with $k$ a positive integer. The space $(X,\tau)$ has the selection property $\textsf{S}_1(\Omega,\Omega)$ if, for every sequence of members of $\Omega$, one can select one member from each so that the selected family belongs to $\Omega$.

Square bracket polarized partition conjecture. If $(X,\tau)$ satisfies

$$\left(\begin{array}{c} \Omega\\ \Omega \end{array}\right) \longrightarrow \left[ \begin{array}{c}\Omega\\ \Omega \end{array}\right]^{1,1}_{k/<3}$$

for any positive integer $k$, then $(X,\tau)$ has the property $\textsf{S}_1(\Omega,\Omega)$.

The conjecture asks whether this square bracket polarized partition relation characterizes the selection property $\textsf{S}_1(\Omega,\Omega)$ for topological spaces. The supplied text does not state whether the assertion is known or unresolved.