Galois representations with big image in the general symplectic group ${\mathrm{GSp}}_4({\mathbb{Z}}_p)$

Let $p$ be an odd prime and $e_p$ be its irregularity index. If $4e_p+8<\frac{p-1}{2}$ we construct a Galois representation with image in the diagonal torus of $\text{op}{GSp}_4(\text{Fp})$ that lifts to a characteristic $0$ representation unramified at all primes $\mathrm{\ell }\mathrm{\ne }p$ with image containing a finite index subgroup of $\text{Symp}$.