Here are some input$/$output examples that show what I mean:

Input: $[1,2,3,4,5,6]$

Output: $[[1],[2,3],[4,5,6]]$

Input: $[1,2,3,4,5,6,7]$

Output: False

In the example $[1,2,3,4,5,6,7],$ the list of lists only has one element in its fourth list:

$[[1],[2,3],[4,5,6],[7]]$

That is NOT equal to the length of the previous list plus one. The last list would have to have four elements to be a valid staircase.