Zagier's modularity conjecture for integral differential-equation solutions

Let $P(t)$ be a non-degenerate quadratic polynomial, and consider the differential equation

$$\bigl(tP(t)y'(t)\bigr)'+(t-\lambda)y(t)=0.$$

An integral solution is a solution at $t=0$ of the form

$$y(t)=1+\sum_{n=1}^{\infty}y_nt^n,$$

with $y_n\in\mathbb{Z}$ for every $n\geq 1$. A solution is modular if it is a modular form expressed locally in terms of a modular function for a monodromy group contained in $\mathrm{SL}_2(\mathbb{R})$. Zagier's modularity conjecture. Any integral solution of the differential equation is modular.

Zagier's computer search found essentially six pairs $(P(t),\lambda)$ with this integrality property, and in every such case the solution is modular. The conjecture asks whether this phenomenon holds for every non-degenerate quadratic $P(t)$ and remains open.