  In this note, we consider the complexity of maintaining the longest
increasing subsequence (LIS) of an array under (i) inserting an element, and
(ii) deleting an element of an array. We show that no algorithm can support
queries and updates in time $\mathcal{O}(n^{1/2-\epsilon})$ and
$\mathcal{O}(n^{1/3-\epsilon})$ for the dynamic LIS problem, for any constant
$\epsilon>0$, when the elements are weighted or the algorithm supports
1D-queries (on subarrays), respectively, assuming the All-Pairs Shortest Paths
(APSP) conjecture or the Online Boolean Matrix-Vector Multiplication (OMv)
conjecture. The main idea in our construction comes from the work of Abboud and
Dahlgaard [FOCS 2016], who proved conditional lower bounds for dynamic planar
graph algorithm. However, this needs to be appropriately adjusted and
translated to obtain an instance of the dynamic LIS problem.
