Constancy of sections in Calabi–Yau Kähler families

Let $p:\mathcal X\to\mathbb D$ be a nonsingular, proper family of Calabi-Yau Kähler manifolds with $K_X\equiv 0$, where $X$ is the central fiber. Let $L\to\mathcal X$ be a holomorphic line bundle, and write $L_0=L|_X$. Assume that for some $m\in\mathbb N^\star$ there is a basis $s_0,\dots,s_N\in H^0(X,mL|_X)$ whose common zero locus is empty. Constancy problem. The function

$$t\to h^0(\mathcal X_t,L|_{\mathcal X_t})$$

is constant. This is posed as an important problem concerning deformation invariance of sections under the stated semiampleness-type hypothesis; the source does not give a resolution.