Let $f:[a,b]\to\mathbb R$ be a Henstock-Kurzweil-integrable function (short: HK-integrable).

Can $f$ always be written as a sum of a Lebesgue-integrable function and a function which has a classical primitive, i.e. are there $f_1\in L^1([a,b])$ and an everywhere differentiable function $F:[a,b]\to\mathbb R$ such that $f=f_1+F'$?

Since every integral function of HK-integrable functions is $ACG_*$, the following question is equivalent:

Can every $ACG_*$-function be written as a sum of an absolutely continuous function and an everywhere differentiable function?

For me it is sufficient to (dis)prove this "only" for HK-integrable functions $f$ which are in $L^1([c,b])$ for each $c\in(a,b)$.

Since HK-integration is brand new to me, I don't really have a feeling whether this is true or not. By reading a couple of books I've gathered a bunch of nice properties of HK-integrable functions, but none of them helped. I would be very grateful if anyone who is more familiar with this type of integration theory can at least have a look at this problem. Thank you very much in advance!