Weakly cofiniteness of local cohomology modules

Let $R$ be a commutative Noetherian ring, $\mathrm{\Phi }$ a system of ideals of $R$ and $I\in \mathrm{\Phi }$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\dim M\le 1$, then the $R$-module ${\text{Ext}}_R^i(R\mathrm{/}I,M)$ is weakly Laskerian, for all $i\ge 0$, if and only if the $R$-module ${\text{Ext}}_R^i(R\mathrm{/}I,M)$ is weakly Laskerian, for $i=0,1$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\text{Ext}}_R^i(R\mathrm{/}I,M)$ is weakly Laskerian for all $i\le t+1$. We prove that if the $R$-module ${\text{lc}}_{\mathrm{\Phi }}^i(M)$ is $\mathrm{F}{\mathrm{D}}_{\le 1}$ for all $i<t$, then ${\text{lc}}_{\mathrm{\Phi }}^i(M)$ is $\mathrm{\Phi }$-weakly cofinite for all $i<t$ and for any $\mathrm{F}{\mathrm{D}}_{\le 0}$ (or minimax) submodule $N$ of ${\text{lc}}_{\mathrm{\Phi }}^t(M)$, the $R$-modules ${\text{Hom}}_R(R\mathrm{/}I,{\text{lc}}_{\mathrm{\Phi }}^t(M)\mathrm{/}N)$ and ${\text{Ext}}_R^1(R\mathrm{/}I,{\text{lc}}_{\mathrm{\Phi }}^t(M)\mathrm{/}N)$ are weakly Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that ${\text{Ext}}_R^j(N,{\text{lc}}_{\mathrm{\Phi }}^i(M))$ and ${\mathrm{T}\mathrm{o}\mathrm{r}}_j^R(N,H_{\mathrm{\Phi }}^i(M))$ are $\mathrm{\Phi }$-weakly cofinite for all $i$ and $j$ whenever $M$ is weakly Laskerian and ${\text{lc}}_{\mathrm{\Phi }}^i(M)$ is ${\rm FD_{\leq 1}}$forall$i$. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.