Dynamic representation conjecture for OCEs in jump-diffusion models

Consider a Markovian claim $X=X^y$ driven by a jump diffusion $Y$, with drift $b$, volatility $\sigma$, jump coefficients $\gamma^{(i)}$, compensator measures $nu^i$, running loss $g$, terminal loss $f$, and OCE loss function $l$ with convex conjugate $l^*$. Let $V$ be the value function and let $\rho$ denote the corresponding OCE risk measure. Dynamic representation conjecture for OCEs in jump-diffusion models. Under suitable assumptions,

$$\rho(X)=V(0,y,1),$$

where $V$ is the minimal viscosity solution of the integro-partial differential equation

$$\begin{aligned} &\partial_tV(s,y,z)+b(s,y)\partial_yV(s,y,z)+\frac{1}{2}\operatorname{tr}\left(\sigma(s,y)\sigma(s,y)'\partial^2_{yy}V(s,y,z)\right)\\ &+\sup_{\theta\in\mathbb{R}^\ell}\Bigl(\sum_{i=1}^\ell\int_{\mathbb{R}^\ell}\Bigl\{V(s,y+\gamma^{(i)}(s,y,\xi),z+z\theta^i)-V(s,y,z)-\partial_yV(s,y,z)\gamma^{(i)}(s,y,\xi)\\ &\qquad-\partial_zV(s,y,z)z\theta^i\Bigr\}\nu^i(d\xi)\Bigr)+\sup_{\beta\in\mathbb{R}^d}\left(\frac{1}{2}z^2|\beta|^2\partial^2_{zz}V+z\,\partial^2_{yz}V\sigma(s,y)\beta\right)+zg(s,y)=0, \end{aligned}$$

with terminal condition

$$V(T,y,z)=f(y)z-l^*(z).$$

The Brownian dynamic representation is established in the paper, whereas in the jump-diffusion setting the authors state that they do not prove this representation; the conjecture is open as far as they know.