Hilbert depth equals Stanley depth for sums of maximal ideals and free modules

Let $R=\mathds{K}[X_1,\ldots,X_n]$ be a polynomial ring in an arbitrary positive number of variables, let $\mathfrak m=(X_1,\ldots,X_n)$ be its homogeneous maximal ideal, and let $\alpha,\beta\in\mathds{N}$. Consider the $\mathds{Z}$-graded module $\mathfrak m^{\oplus\alpha}\oplus R^\beta$. The Hilbert–Stanley depth conjecture. The $\mathds{Z}$-graded Hilbert depth of $\mathfrak m^{\oplus\alpha}\oplus R^\beta$ and its Stanley depth coincide. The conjecture is presented as a strengthening of a preceding proposition and is based on several examples; the supplied source gives no further evidence of resolution.