Creating a W state for $2^k$ qubits

This question is available in Quantum Katas here

In Task 2.6 of this notebook, we are required to create the W State for $2^k$ qubits.

Input: $?=2^?$ qubits in the |0…0⟩ state.

Goal: Change the state of the qubits to the W state – an equal superposition of ? basis states on ? qubits which have Hamming weight of 1.

For example, for $?=4$ the required state is $frac{1}{2}(|1000⟩+|0100⟩+|0010⟩+|0001⟩)$

And since this problem is taken from Katas, it came with a solution which is as follows:
s

operation WState_PowerOfTwo (qs : Qubit[]) : Unit is Adj+Ctl {
    let N = Length(qs);

    if (N == 1) {
        // base of recursion: |1⟩
        X(qs[0]);
    } else {
        let K = N / 2;
        using (anc = Qubit()) {
            H(anc);
            
            (ControlledOnInt(0, WState_PowerOfTwo))([anc], qs[0 .. K - 1]);
            (ControlledOnInt(1, WState_PowerOfTwo))([anc], qs[K .. N - 1]);

            for (i in K .. N - 1) {
                CNOT(qs[i], anc);
            }
        }
    }
}

While there is absolutely nothing wrong with the proposed answer, I was trying to solve this task without using an ancilla qubit, here’s my approach to this question:

operation WState_PowerOfTwo (qs : Qubit[]) : Unit {
    let length_qs = Length(qs);
    if (length_qs == 1){
        X(qs[0]);
    }
    else{
    H(qs[0]);
    
    for(i in 1..length_qs-1){
        if(i != length_qs-1){
            for (j in 0..i-1){
                X(qs[j]);
            }
            Controlled H(qs[0..i-1], qs[i]);
            for (j in 0..i-1){
                X(qs[j]);
            }
        }
        else{
            for (j in 0..i-1){
                X(qs[j]);
            }
            Controlled X(qs[0..i-1], qs[i]);
            for (j in 0..i-1){
                X(qs[j]);
            }
        }
    }
}
}

This logics works fine till N=2 but it shows the following error when testing for hidden cases:

The desired state for N = 1
# wave function for qubits with ids (least to most significant): 0
∣0❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
∣1❭:     1.000000 +  0.000000 i  ==     ******************** [ 1.000000 ]     --- [  0.00000 rad ]
The actual state:
# wave function for qubits with ids (least to most significant): 0
∣0❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
∣1❭:     1.000000 +  0.000000 i  ==     ******************** [ 1.000000 ]     --- [  0.00000 rad ]
Test case passed
The desired state for N = 2
# wave function for qubits with ids (least to most significant): 0;1
∣0❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
∣1❭:     0.707107 +  0.000000 i  ==     ***********          [ 0.500000 ]     --- [  0.00000 rad ]
∣2❭:     0.707107 +  0.000000 i  ==     ***********          [ 0.500000 ]     --- [  0.00000 rad ]
∣3❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
The actual state:
# wave function for qubits with ids (least to most significant): 0;1
∣0❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
∣1❭:     0.707107 +  0.000000 i  ==     ***********          [ 0.500000 ]     --- [  0.00000 rad ]
∣2❭:     0.707107 +  0.000000 i  ==     ***********          [ 0.500000 ]     --- [  0.00000 rad ]
∣3❭:     0.000000 +  0.000000 i  ==                          [ 0.000000 ]                   
Test case passed
Testing on hidden test cases...
Released qubits are not in zero state.
Try again!

I cannot find a root for this problem, is there any error in my logic or am I missing something here?