Are these bra-ket manipulations correct?

I’m reading an old paper ("Wigner’s Function and Other Distributions in Mock Phase Spaces," Balazs and Jennings, Phys. Rep. 104(6), 1984), and came across the following statement (in which $hat{q}$ and $hat{p}$ are a pair of generalized canonical operators and $hat{rho}$ is some density operator):

We may also evaluate $text{Tr}[delta(hat{q}-q’)delta(hat{p}-p’)hat{rho}]$
or $text{Tr}[delta(hat{p}-p’)delta(hat{q}-q’)hat{rho}]$ … These
quantities, however, are not the same, are not symmetrical in
$hat{p}$ and $hat{q}$ and are not positive everywhere. Furthermore,
they will be complex.

It seemed intuitive to me that these traces could be different in general (since $hat{q}$ and $hat{p}$ don’t commute), but it wasn’t so clear that this quantity could be negative or imaginary even though clearly the wavefunction in position or momentum basis may be.

So I tried seeing this by evaluating a bit. Let $hat{rho} = left|psirightrangle leftlangle psi right|$ and $left|psirightrangle = int dq , psi(q)left|qrightrangle$ in the position basis. Then I thought to do the trace in the position basis, i.e.

$$ begin{aligned}
text{Tr}[delta(hat{q}-q’)delta(hat{p}-p’)hat{rho}] &= int dq” leftlangle q” | delta(hat{q}-q’)delta(hat{p}-p’)hat{rho} | q” rightrangle
&= int dq” int dq leftlangle q”|delta(hat{q}-q’)delta(hat{p}-p’)|q rightrangle langle q|q” rangle |psi(q)|^2
&= int dq” int dq leftlangle q”|delta(hat{q}-q’)delta(hat{p}-p’)|q” rightrangle |psi(q)|^2
&= langle q’ | delta(hat{p}-p’) | q’ rangle |psi(q)|^2
&= int dp” langle q’|delta(hat{p}-p’)|p” rangle langle p”|q’ rangle |psi(q)|^2
&= langle q’|p’ rangle langle p’|q’ rangle |psi(q)|^2
&= |langle q’|p’ rangle|^2 |psi(q)|^2
end{aligned} $$

which is always real and positive, contrary to the statement of the paper. So I have to believe I’ve done something wrong. Any help in pointing out my error would be greatly appreciated.