Conjecture that morphisms to normal varieties are lambda-uniform

Let $\mathbb{F}_q$ be a finite field, and let $Y_0$ and $Z_0$ be varieties over $\mathbb{F}_q$. A morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform if, for every untwisted compatible system $\underline{\mathcal{V}_0}$ over $Y_0$, its pullback $f_0^*\underline{\mathcal{V}_0}$ is $\lambda$-uniform. Lambda-uniformity conjecture. If $Y_0$ is normal, every morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform. The conjecture is proposed as the central strengthening studied in the paper. The preceding theorem establishes strong $\lambda$-uniformity for compatible systems on normal varieties, but the assertion for arbitrary source varieties and morphisms remains open.