Hasse–Arf theorem for arithmetic conductors

Let $k$ be a complete discretely valued field of equal characteristic $p$, and let $G_k$ be its absolute Galois group. Let $\rho$ be a representation of $G_k$ with finite local monodromy. Define the arithmetic Artin and Swan conductors by

$$\mathrm{Art}_{ar}(\rho)=\sum_{a\in\mathbb{Q}_{\geq 0}}a\cdot\dim\left(V_\rho^{\mathrm{Fil}^{a+}G_k}/V_\rho^{\mathrm{Fil}^aG_k}\right),$$ $$\mathrm{Swan}_{ar}(\rho)=\sum_{a\in\mathbb{Q}_{\geq 0}}a\cdot\dim\left(V_\rho^{\mathrm{Fil}_{\mathrm{log}}^{a+}G_k}/V_\rho^{\mathrm{Fil}_{\mathrm{log}}^aG_k}\right).$$

Hasse–Arf theorem. The arithmetic conductors are nonnegative integers:

$$\mathrm{Art}_{ar}(\rho)\in\mathbb{Z}_{\geq 0},\qquad \mathrm{Swan}_{ar}(\rho)\in\mathbb{Z}_{\geq 0}.$$

This is the equal-characteristic Hasse–Arf integrality assertion for the arithmetic conductors associated with the ramification filtrations. The supplied text does not state whether this theorem is being proved, recalled, or treated as open.