maximum likelihood estimation of the discrete normal distribution's parameters

Question

If we consider the discrete normal distribution (https://www.tandfonline.com/doi/abs/10.1081/STA-120023256) whose pmf is equal for any integer value $x$ to $$p(x)=\Phi((x+1-\mu)/\sigma)-\Phi((x-\mu)/\sigma)$$ (with $\Phi$ the cdf of a standard normal random variable, $\mu\in\mathbb{R}$, $\sigma>0$) and I want to compute the MLEs of $\mu$ and $\sigma$ by equating to zero the two first-order partial derivatives of $\ell(\mu,\sigma;x_1,\dots,x_n)=\sum_{i=1}^n \log p(x_i;\mu,\sigma)$... I wonder whether there is a closed-form solution for the MLEs. It looks like the MLE of $\mu$ is $\bar{x}+0.5$, but I'm not able to derive it from the two normal equations.