An identity involving products of three binomial coefficients

Let $l,m$ and $n$ be nonnegative integers. We establish the following new identity: $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}.$$ In the case $l=n$ this confirms a conjecture of V.J.W. Guo; because the right hand side becomes $\binom{2m/3}{m/3}\binom{m/3}{n-m/3}$ if $3|m$, and 0 otherwise.