On algebraic cycles on fibre products of non-isotrivial families of regular surfaces with geometric genus 1

Let $\pi_k:X_k\to C\,\,\,(k = 1, 2)$ be a projective family of surfaces (possibly with degenerations) over a smooth projective curve $C$. Assume that the discriminant loci $\Delta_k=\{\delta\in C\,\,\vert\,\,\operatorname{Sing}(X_{k\delta})\neq\varnothing\} \quad (k = 1, 2)$ are disjoint, $h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0$ for any smooth fibre $X_{ks}$ and the period map associated with the variation of Hodge structures $R^2\pi'_{k\ast}\mathbb{Q}$ (where $\pi'_k:X'_k\to C\setminus\Delta_k$ is a smooth part of the morphism $\pi_k$), is non-constant. If for generic geometric fibres $X_{1s}$ and $X_{2s}$ the following conditions hold:
(i) $b_2(X_{1s})-\operatorname{rank} \operatorname{NS}(X_{1s})$ is an odd integer;
(ii) $b_2(X_{1s})-\operatorname{rank}\operatorname{NS}(X_{1s})\neq b_2(X_{2s})-\operatorname{rank} \operatorname{NS}(X_{2s})$,
then for any smooth projective model $X$ of the fibre product $X_1\times_CX_2$ the Hodge conjecture on algebraic cycles is true.
If, besides, the morphisms $\pi_k$ are smooth, $p_k=b_2(X_{ks}) -\operatorname{rank} \operatorname{NS}(X_{ks}) \,\,\,(k = 1,2)$ are odd prime numbers and $p_1\neq p_2$, then for $X_1\times_CX_2$ and for the fibre square $X_1\times_CX_1$ the Hodge conjecture and the Grothendieck standard conjecture on algebraicity of operators $\ast$ and $\Lambda$ of Hodge theory hold as well.
This result yields new examples of smooth projective 5-dimensional varieties satisfying the Hodge and the Grothendieck conjectures, because one can take as smooth fibres of the morphism $\pi_k:X_k\to C$ some $K3$ surfaces, minimal regular surfaces of general type (of Kodaira dimension $\varkappa=2$) with geometric genus $1$ belonging to one of the following types: (a) surfaces with $K^2\leq 2$; (b) surfaces with $3\leq K^2\leq 8$, whose moduli are in the same component of the space of moduli as Todorov surface; (c) surfaces with $K^2 = 3$ with torsion of the Picard group $\mathbb{Z}/3\mathbb{Z}$.