We address the open problem posed by B. Dyda for the critical case sp=1 by deriving a fractional Hardy inequality with a boundary singularity for sp=1 using an optimal logarithmic weight function. Furthermore, we extend our findings to the case p=1 by establishing the corresponding fractional Hardy inequality with a boundary singularity on bounded Lipschitz domains, incorporating logarithmic corrections. Utilising the result by Brezis, Bourgain, and Mironescu on the limiting behavior of fractional Sobolev spaces as s approaches 1, we obtain a Hardy inequality with a boundary singularity for functions of bounded variation on bounded Lipschitz domains.