Is the primality problem with unary input NLOGSPACE-Hard? Consider the language $L=\{1^n : n \text{ is prime}\}$. Is this language NLOGSPACE-Hard?

The motivation for this question is that $L$ is a good candidate for reducing to other languages related to my automata research.
This language is in $\mathsf{LOGSPACE}$ via trial division. It is also known logarithmic space is neccessary ([1]). For a generalization to sparse sets, see bounded language complete for NSPACE(log n)?. For hardness in binary case, see Are the problems PRIMES, FACTORING known to be P-hard?.

[1] J. Hartmanis, L. Berman, On tape bounds for single letter alphabet language processing. Theoretical Computer Science vol. 3 http://www.sciencedirect.com/science/article/pii/0304397576900244