Flatness on the formal plane from flatness on lines through the origin?

Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of the form $t=ah$ for $a\in \mathbb C$.

Can we conclude that $\mathcal F$ is flat?

For coherent sheaves, I think I can see an argument by comparing dimensions of residues (if the dimension of the residue at the closed point is the same as at the generic points of infinitely lines, it also must be the same as that at the generic point), but of course, that doesn't make sense for quasi-coherent sheaves.

Is there some better argument, or can the behavior of these modules be more pathological than I realize? I'd really appreciate a pointer toward a reference, as well (or at least guidance on where to look for it).

EDIT: I realized I should also have a homogeneity hypothesis, and thus have asked a new question.

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1 Answer

On $X=\mathrm{Spec}\,\mathbb{C}[[t,h]]$, choose an irreducible curve of degree $\geq2$ (e.g. $t^2=h$) and let $\eta$ be its generic point, $j:\eta\to X$ the inclusion. Then $\mathcal{F}:=j_*\mathcal{O}_\eta$ is quasicoherent and zero on every line through the origin, but not flat. (In terms of $\mathbb{C}[[t,h]]$-modules, it is just the fraction field of $\mathbb{C}[[t,h]]/(f)$ where $f$ is the equation of the curve).

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