Zagier-type conjecture for integral differential-equation solutions

Consider the differential equation

$$\left( b_3t^3 + b_2t^2 + b_1t +b_0 \right) F(t) + \left((c_5t^5+c_4t^4+c_3t^3+c_2t^2 -t)F'(t)\right)'=0.$$

An integral solution means a solution of this equation whose coefficients are integral. Zagier-type conjecture. Any integral solution to this differential equation corresponds to the solution of a Picard–Fuchs equation about a singular fiber. This claim is presented as evidence from a brute-force search and is described as a variant of one of Zagier's conjectures. The precise correspondence and its general scope are not further specified in the supplied text.