Here we work with the formal power series ring $K[[x]]$ where $K$ is a field of characteristic zero. The Hadamard product of two such power series is the coefficient wise product: thus if $f= \sum a_n x^n$ and $g = \sum b_n x^n$ are two power series then $f * g$ is defined to be $$f* g := \sum a_n b_n x^n.$$ Notice that the function $(1-x)^{-1}$ is the identity for this operation. If the coefficients of $f$ are all nonzero, then it has a "Hadamard inverse" by taking the reciprocals of its coefficients and forming a new power series: $$\bar{f} := \sum \frac{1}{a_n} x^n.$$
Under Hadamard multiplication and usual addition, this is just the ring of sequences in $K$, which a priori has nothing to do with the usual product of power series. But here is the remarkable theorem on rationality of Hadamard quotients, connecting the Hadamard product to the usual product:
Theorem: Suppose that $f ,g$ are rational power series with $g$ Hadamard-invertible, and $f*\bar{g}$ has coefficients in a finitely-generated ring. Then $f*\bar{g}$ is also rational.
This can be viewed as a rationality test: if all the coefficients lay in a finitely-generated ring (which is the case for a D-finite series), and you can write the coefficients as a quotient of two linear recurrences, then the power series is rational.
I'm wondering if there is a similar Hadamard quotient theorem when "rational" is replaced by "D-finite". A power series $f$ is D-finite if it satisfies a linear differential equation with polynomial coefficients; equivalently, its coefficient sequence $(a_n)$ satisfies a linear recurrence whose coefficients are rational functions in $n$ (in which case $(a_n)$ is called P-recursive).
A quick google search finds a conjecture of Dimitrov, that if $f,g$ are D-finite with finite "height" and $f*\bar{g}$ also has finite height, then $f*\bar{g}$ is also D-finite. What is known about this conjecture? Unfortunately google and citation searches have not helped me.
Thanks!