Parameterized dynamic data structure for Split Completion

We design a randomized data structure that, for a fully dynamic graph $G$ updated by edge insertions and deletions and integers $k,d$ fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add $k$ edges to $G$ to obtain a split graph. The data structure can be initialized on an edgeless $n$-vertex graph in time $n\cdot (kd\cdot \log n{)}^{\mathcal{O}(1)}$, and the amortized time complexity of an update is $5^k\cdot (kd\cdot \log n{)}^{\mathcal{O}(1)}$. The answer provided by the data structure is correct with probability $1-\mathcal{O}(n^{-d})$.