Additivity conjecture for generalized Jordan derivations on triangular rings

Let $A$ and $B$ be rings and let $\mathcal{T}$ be a triangular ring satisfying the following conditions: for $a\in A$, $aA=0$ implies $a=0$; and for $b\in B$, $bB=0$ implies $b=0$. Let $\delta:\mathcal{T}\rightarrow\mathcal{T}$ be a map and let $\tau:\mathcal{T}\rightarrow\mathcal{T}$ satisfy

$$\tau(ab+ba)=\tau(a)b+a\tau(b)+\tau(b)a+b\tau(a)$$

for all $a,b\in\mathcal{T}$. If

$$\delta(ab+ba)=\delta(a)b+a\tau(b)+\delta(b)a+b\tau(a)$$

for all $a,b\in\mathcal{T}$, then $\delta$ is additive. Moreover, if $\mathcal{T}$ is $2$-torsion free, then $\delta$ is a generalized Jordan derivation. This conjecture removes condition (iii) from the preceding theorem; the example given shows that omitting that condition does not alter the expected conclusion, but the source provides no resolution of the conjecture.