Existence of a power difference set with index four satisfying the square condition

Determine whether any power difference set with \(\ell=4\) exists for a prime power \(q\) satisfying \(\ell(k-\lambda)=q-k\) equal to a perfect square.

Background

Theorem 4.13 establishes that if a power difference set has $\ell(k-\lambda)=q-k$ a perfect square, then either $\ell$ is divisible by $4$, or $\ell$ is a sum of two odd squares. The paper gives examples for $\ell=2$ and $\ell=8$.

Although the theorem’s condition is satisfied in examples with $\ell=2$ and $\ell=8$, the paper states that no example is known for $\ell=4$, leaving this existence question unresolved.