ca.analysis and odes - Trigonometry related to Rogers--Ramanujan identities

For integers $nge2$ and $kge2$, fix the notation
$$
[m]=sinfrac{pi m}{nk+1} quadtext{and}quad
[m]!=[1][2]dots[m], qquad minmathbb Z_{>0}.
$$
Consider the following trigonometric numbers:
$$
a_i=frac{[i+k-2]![n-i+k-2]!}{[k-2]![n+k-2]!},
qquad i=1,2,dots,n-1.
$$
Is it possible for any $n$ to express the quantities
$$
A_j=1-frac{[k-1]cdot [n+k-1]}{[j+k-1]cdot [n-j+k-1]},
qquad j=1,2,dots,n-1,
$$
as a product/quotient of terms of the form
$(1-text{product of some }a_i)$? If not (for $nge4$),
is it possible to prove that?

The affirmative answer is known for $n=2$ and $n=3$.
Namely, if $n=2$ so that we have only one $a_1$ and one $A_1$, then
$$
A_1=1-a_1.
$$
If $n=3$, then
$$
A_j=(1-a_j)(1-a_1a_2) quadtext{for } j=1,2.
$$
(The last formula is a nice trigonometric identity, by the way.)

My question is motivated (in a very sophisticated way) by a recent
question on Rogers--Ramanujan identities.
The latter one reminded me about the problem of possible $mathfrak{sl}_n$ generalizations of RRs
in their classical form "a $q$-sum"="a $q$-product". The only cases $n=2$ and $n=3$ are known;
these are the Andrews--Gordon identities and the Andrews--Schilling--Warnaar identities
(see [S. Ole Warnaar, Adv. in Math. 200 (2006) 403--434]).
An indirect implication of such identities is the family of (highly nontrivial)
numerical identities for the dilogarithm function; these come as the limit $qto1$ specialisation
and some multivariate asymptotics. The trigonometric identities above come into
play from these considerations for $n=2$ and $n=3$; any answer for $n>3$ can shed
some light on the existence of new RRs.