  We present a candidate of a vector space basis for the algebra
$\mathcal{O}(S_q^{4n-1})$ of the quantum symplectic sphere for every $n\geq 1$.
The algebra $\mathcal{O}(S_q^{4n-1})$ is defined as a certain subalgebra of the
quantum symplectic group $\mathcal{O}(SP_q(2n))$. A non-trivial application of
the Diamond Lemma is used to construct the vector space basis and the
conjecture is supported by computer experiments for $n=1,2,...,8$.
