Garg and Sharma's permutation trinomial conjecture over fields of characteristic two

Let $m$ be a positive integer and define

$$g(x):=x^5+x^{2^m+4}+x^{5\times 2^m}\in\mathbb{F}_{2^{2m}}[x].$$

A polynomial over a finite field is a permutation polynomial if it induces a one-to-one map from the field to itself. Garg–Sharma's conjecture. The polynomial $g(x)$ is a permutation trinomial over $\mathbb{F}_{2^{2m}}$ if and only if

$$m\equiv 2\pmod{4}.$$

This conjecture concerns an explicit family of permutation trinomials in even characteristic. Its status is not established by the supplied source context.