こちらの問題を Java で解きました。
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1 時間では解き切れなかったので敗北です。
package org.example;

import java.util.ArrayList;
import java.util.Arrays;
import java.util.Comparator;
import java.util.List;
import java.util.stream.Collectors;
import java.util.stream.Stream;

public class Solver {

  public String solve(String src) {
    var squares = new ArrayList<Square>();
    Arrays.stream(src.split(",")).map(Integer::parseInt).forEach(length -> {
      var nextX = Integer.MAX_VALUE / 2;
      var nextY = Integer.MAX_VALUE / 2;
      for (int x : xs(squares)) {
        for (int y : ys(squares)) {
          var square = new Square(new Point(x, y), length);
          if (squares.stream().allMatch(s -> s.isNotOverlapped(square))) {
            // できるだけ下 ( x + y が小さい ) または同じ高さ ( x + y が同じ ) でできるだけ左 ( y が大きい ) が見つかったか判定
            boolean b = (x + y < nextX + nextY) || (x + y == nextX + nextY) && (nextY < y);
            if (b) {
              nextX = x;
              nextY = y;
            }
          }
        }
      }
      squares.add(new Square(new Point(nextX, nextY), length));
    });
    Square maxSquare = squares.stream().max(Comparator.comparingInt(Square::length)).get();
    return squares
        .stream()
        .filter(maxSquare::isNeighbor)
        .mapToInt(Square::length)
        .sorted()
        .mapToObj(String::valueOf)
        .collect(Collectors.joining(","));
  }

  private List<Integer> xs(List<Square> squares) {
    return Stream.concat(
        Stream.of(0),
        squares.stream().map(square -> square.point.x + square.length)
    ).distinct().toList();
  }

  private List<Integer> ys(List<Square> squares) {
    return Stream.concat(
        Stream.of(0),
        squares.stream().map(square -> square.point.y + square.length)
    ).distinct().toList();
  }

  private record Point(int x, int y) {

  }

  private record Square(Point point, int length) {

    public boolean isNotOverlapped(Square other) {
      return (point.x + length <= other.point.x) || (other.point.x + other.length <= point.x) || (
          point.y + length <= other.point.y) || (other.point.y + other.length <= point.y);
    }

    public boolean isNeighbor(Square other) {
      if ((point.x + length == other.point.x) || (point.x == other.point.x + other.length)) {
        return (other.point.y < point.y + length) && (point.y < other.point.y + other.length);
      }
      if ((point.y + length == other.point.y) || (point.y == other.point.y + other.length)) {
        return (point.x < other.point.x + other.length) && (other.point.x < point.x + length);
      }
      return false;
    }
  }
}