A note on balancing binomial coefficients

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\left(\genfrac{}{}{0px}{}{1}{5}\right)+\left(\genfrac{}{}{0px}{}{2}{5}\right)+\mathrm{.}\mathrm{.}\mathrm{.}+\left(\genfrac{}{}{0px}{}{x-1}{5}\right)=\left(\genfrac{}{}{0px}{}{x+1}{5}\right)+\mathrm{.}\mathrm{.}\mathrm{.}+\left(\genfrac{}{}{0px}{}{y}{5}\right)$$ where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic logarithms.