The sharp upper bound for the Swan-exponent difference in residue characteristic two

Let $p$ be the residue characteristic, and let $\rho$ be a Galois representation. The Swan-exponent difference is

$$\operatorname{Sw}(\operatorname{Sym}^{2}\rho)-\operatorname{Sw}(\wedge^{2}\rho).$$

Sharp upper-bound conjecture. Assume $p=2$. For any Galois representation $\rho$, one has

$$\operatorname{Sw}(\operatorname{Sym}^{2}\rho)-\operatorname{Sw}(\wedge^{2}\rho)\leq\operatorname{Sw}(\rho).$$

This would improve the established upper bound by a factor of two in residue characteristic two. The source further indicates that the proposed inequality is expected to be sharp, but does not state a resolution.