Arithmetic support and monodromy

In my talk on Arbeitstagung 2017 I introduced a strange series with rational coefficients related to p-curvatures of certain family of algebraic linear differential equations (Heun equations). This series has zero radius of convergence p-adically for any prime p, which seems to be a new phenomenon, and it is extremely hard to calculate (e.g. using first 40000 primes one can get only first 250 coefficients of the series).
 I then claimed that this series is given by the determinant of the logarithm of the monodromy of the equation, understood as a function of the parameter, and indicated a possible approach to the proof.
In reality, my arguments were not complete, and  3 different attempts to prove the conjectured identity (by Alexander Odessky, Don Zagier, and myself), all ended in failure.

Two months ago, together with A.Odessky, we finally obtained a proof by a calculation-free argument.
I'll explain the proof, and, if the time permits, will put it into a general context.