I think so. You must find length of max increasing subsequence in array B. Array B = A + A, where A is array that we were given. It can be done in O(nlog(n))

int t;
cin >> t;
while (t > 0){
  int n;
  cin >> n;
  int a[2*n];
  for (int i = 0; i < n; i++){
    cin >> a[i];
    a[i + n] = a[i];
  }
  vector < int > seq(2*n + 1, 2e9);
  seq[0] = -1;
  int ans = 0;
  for (int i = 0; i < 2*n; i++){
    int p = (int)(upper_bound(seq.begin(), seq.end(), a[i]) - seq.begin());
    if (a[i] > seq[q - 1]){
      seq[p] = a[i];
      ans = max(ans, p);
    }
  }
  cout << ans << endl;
  t--;
}

Clearly you don't have enough time to enumerate all those sequences, so that direction won't lead very far. Note that instead of thinking that Catwoman moves from right to left in an increasing sequence, the problem doesn't change if she moves from left to right in a decreasing sequence.

Why is this helpful? Because now Batman and Catwoman are moving in the same direction, which should let you use a really well-known technique to construct both of their paths at the same time.

we can see that its finding disjoint increasing subsequence and longest decreasing subsequence used simple DP

dp[pos][x][y] = the longest increasing subsequence and decreasing subsequence, using elements 1..pos, and the last index of longest increasing is x, while the last index of longest decreasing is y

then dp[pos][x][y] is the max of the following:

dp[pos+1][x][y] //not take the pos element
dp[pos+1][pos][y] + 1 //take the pos element as the next element in increasing (a[pos] > a[x])
dp[pos+1][x][pos] + 1 //take the pos element as the next element in decreasing (a[pos < a[y])

Found the error, fixed