= Solution
Define the <salinity> contrast $\Delta S=S_1-S_2$, positive when the warm box is saltier. The linear <equation of state> gives the cold-minus-warm <density> contrast and signed <volume flux>
$$
\frac{\rho_2-\rho_1}{\rho_0}=\alpha\Delta T-\beta\Delta S,\qquad
q=k(\alpha\Delta T-\beta\Delta S).
$$
Here $q>0$ represents warm upper water travelling poleward and cold deep water returning equatorward. The $\beta$ in this question is the haline <density> coefficient, unrelated to the planetary <vorticity> gradient in Question 1. The imposed heating and cooling are taken to maintain $\Delta T$.
For the positive circulation shown in the source diagram, the exact stationary <salt budget> of the cold box is
$$
qS_1-(q+E)S_2=0,
\qquad q\Delta S=ES_2.
$$
The precipitation adds no salt. Replacing the salinity multiplying the small compensating <volume flux> by a reference $S_0$ gives $q\Delta S\simeq ES_0$. When circulation reverses, the upstream box for the exchange reverses too: the homogenizing exchange is $|q|$, not $q$. In the same reference-salinity approximation the two stationary <salt budgets> reduce to
$$
\boxed{|q|\Delta S=ES_0.}
$$
For example, when $q=-r<0$ and $r>E$, the exact cold-box balance is $(r-E)S_1-rS_2=0$, giving $r\Delta S=ES_1$ and the same leading approximation.
Substitute $\Delta S=ES_0/|q|$ into the <density>-driven <volume flux> and multiply by $|q|$:
$$
\boxed{q|q|-A|q|+BE=0,\qquad
A=k\alpha\Delta T,\qquad B=k\beta S_0.}
$$
Both $A$ and $B$ have the units of <volume flux>, so every term has units of squared <volume flux>. The absolute value expresses the fact that transport in either direction reduces the <salinity> contrast, while the <density> contrast sets its direction.
The requested equation is thus the reference-salinity, or <virtual salt flux>, version of the <two-box overturning model>. If the finite compensating flow is retained exactly with unequal $S_1,S_2$, one has the appropriate upstream salinity on the right-hand side instead of exactly $S_0$. The approximation assumes $|\Delta S|/S_0\ll1$ and $E/|q|\ll1$. Also, for $E>0$, $q=0$ cannot be a steady solution: the imposed freshwater transport would build an uncompensated <salinity> contrast.
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