Harris's conjecture on Noether–Lefschetz loci for octic surfaces

Let $X_0: x_0^8+x_1^8+x_2^8+x_3^8=0$ be the Fermat octic, and let $C_1$ and $C_2$ be respectively a line, namely a complete intersection of type $(1,1)$, and a complete intersection of type $(3,3)$ inside $X_0$, obtained by canonical factorization of $x_0^8+x_1^8$ and $x_2^8+x_3^8$. Assume that $C_1$ does not intersect $C_2$. For $r\in\mathbb{Q}$ with $r\ne 0$, set

$$\delta_0=[C_1]+r[C_2]\in H^2(X_0,\mathbb{Q}).$$

Let $\delta_t$ be its parallel transport to $H^2(X_t,\mathbb{Q})$ for $t\in({\sf T},0)$, where ${\sf T}$ is the full parameter space of smooth octic surfaces, and let $V_{[C_1]+r[C_2]}\subset{\sf T}$ be the local Noether–Lefschetz locus parameterizing those $X_t$ for which $\delta_t$ remains of type $(1,1)$. Let $V_{C_i}$ be the local branch parameterizing deformations of $(X_0,C_i)$, with $C_1$ a line and $C_2$ a complete intersection of type $(3,3)$. Harris's conjecture. The local Noether–Lefschetz loci $V_{[C_1]+r[C_2]}$, for $r\in\mathbb{Q}$ and $r\ne0$, are smooth, distinct analytic spaces of codimension $31$, passing through the common $32$-codimensional subvariety $V_{C_1}\cap V_{C_2}$. In particular, their underlying analytic varieties are distinct and have codimension less than $35$. The conjecture concerns the local geometry and distinctness of Noether–Lefschetz loci associated with rational combinations of the classes of the two curves; the paper presents strong evidence for these claims, while the supplied text gives no resolution.