Does there exist an odd natural number $N = xy$ satisfying $D(x)D(y) = 2s(x)s(y)$?

Question

Preamble: I apologize in advance if what I am asking for in this question, I could get an answer easily so myself, for example by coding a short Mathematica script. It is just that I have not yet studied the rudiments of Mathematica (or even Python) scripting, so I am not there yet.

This question is a follow-up to this earlier MSE question.

Let $\sigma(z)$ be the sum of the divisors of $z \in \mathbb{N}$. Denote the deficiency of $z$ by $D(z) := 2z - \sigma(z)$ and the sum of the aliquot parts of $z$ by $s(z) := \sigma(z) - z$.

Here is my question:

Does there exist an odd natural number $N = xy$ satisfying $D(x)D(y) = 2s(x)s(y)$?

Answer 1

The answer is "YES" for the Descartes spoof $$\mathscr{D} = 198585576189 = {22021}\cdot{3003}^2,$$ where we set $x = 22021$ and $y = {3003}^2$. (That is, if we pretend that $x = 22021$ is prime.)

We "obtain" $$D(x)D(y) = 18034380 = 2s(x)s(y).$$