Strict dyadic Swan-exponent inequality for irreducible orthogonal representations

Let $G$ be the relevant local Galois group, let $p$ be its residue characteristic, and let $\rho$ be a finite-dimensional Galois representation. Write $\operatorname{Sw}(\rho)$ for its Swan conductor, and write $\operatorname{Sym}^{2}\rho$ and $\wedge^{2}\rho$ for its symmetric and exterior square representations. A representation is orthogonal if it admits a nondegenerate $G$-invariant symmetric bilinear form, and it is tame if its wild inertia subgroup acts trivially.

Strict inequality conjecture. Assume $p=2$. For any irreducible orthogonal Galois representation $\rho$ of $G$ which is not tame, one has

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

This is presented as a further expected consequence supported by the paper's induction argument and is not proved in general.