Conjecture B on curvewise bounded ramification and logarithmic conductors

Keep the hypotheses of Conjecture A: $k$ is perfect, $X/k$ is normal, $j:U\hookrightarrow X$ is a $k$-smooth open immersion with complement supporting an effective Cartier divisor, $D$ is an effective Cartier divisor supported on $X\setminus U$, and $\mathcal{F}$ is a smooth $\ell$-adic sheaf of finite rank on $U$. Say that the ramification of $\mathcal F$ is bounded by $D$ if, for every $C\in\mathrm{Cu}(U)$, the divisor $\mathrm{Sw}(\phi^*\mathcal F)$ on the normalisation $\bar C$ is at most $\bar\phi^*D$, and write this as $\mathrm{Sw}(\mathcal F)\leq D$. Conjecture B. The following are equivalent:

1. $\mathrm{Sw}(\mathcal F)\leq D$.
2. For every open immersion $j':U\subset X'$ over $k$ such that $X'\setminus U$ is a simple normal crossing divisor, and every morphism $h:X'\to X$ extending $j:U\to X$, one has

$$\mathrm{Sw}_{\xi'}(h^*j_!\mathcal F)_{\mathrm{log}}\leq m_{\xi'}(h^*D)$$

for each generic point $\xi'\in D':=X'\setminus U$. This gives an intrinsic logarithmic-conductor criterion equivalent to detecting bounded ramification on all curves, and is presented as a conjectural extension of the rank-one theory; the source does not specify its resolution status.