Experimentally, the sequence converges to $1,$ at the logarithmic rate suggested in Fedor's answer. Here is the graph for the first 20000 numbers:

[![enter image description here][1]][1]


Now, when we fit the actual $ds_{10}(3^m),$ we get the following suggestive graph:

[![enter image description here][2]][2]

Whose slope is pretty close to Fedor's "probabilistic" value. *HOWEVER* convergence is slow - empirically, Fedor's $o(1)$ term is actually of order $1/\log m.$

  [1]: https://i.sstatic.net/W68DP.png
  [2]: https://i.sstatic.net/HDTN2.png