The surjectivity conjecture for cohomology of smoothings

Let $X_0$ be a normal crossing variety with smoothing $X_t$, and let $X$ denote the associated total space. Write $H^k(-,\mathbb Z)_f$ for the torsion-free quotient of integral cohomology, and suppose that $h^{2,0}(X_t)=0$. Let $RG^2(X_0,\mathbb Z)$ and $RG^{2n-2}(X_0,\mathbb Z)$ be the relevant cohomology groups of the normal crossing central fibre, equipped with their natural pairing. Condition

is the assertion that this pairing is unimodular. **The surjectivity conjecture.** When $h^{2,0}(X_t)=0$, Condition

holds; equivalently, the maps

$$H^2(X,\mathbb Z)_f\longrightarrow H^2(X_t,\mathbb Z)_f$$

and

$$H^{2n-2}(X,\mathbb Z)_f\longrightarrow H^{2n-2}(X_t,\mathbb Z)_f$$

are surjective. The conjecture would make the integral cohomology of the smoothing, modulo torsion, computable from the corresponding groups of the normal crossing central fibre. The paper notes that all examples considered satisfy the condition, but gives no proof in general.