My quickselect returns kth value, rather than kth smallest value

Question

I'm trying to implement Quickselect, which should find the kth smallest element in an array of ints. However, the function findKthSmallest() is returning the kth value, rather than the kth smallest. I can't figure out why this is happening - I tried a lot of small changes, and refactored and rewrote it a couple times now. It seems like it should be really simple, but I don't know. Here is my C++ code:

#include <iostream>
#include <vector>
#include <random>
#include <string>
using namespace std;

//note: when i put an 'I' at the end of a variable name, it stands for Index. (and V stands for Value)

string vecToStr(vector<int>);

class KSE {//KSE stands for Kth Smallest Element
public:
    KSE(vector<int> S, int K);
    vector<int> set;
    int k;
    int kthSmallest;
private:
    int findKthSmallest();
//  int findKthSmallest(int leftI, int rightI);
    int partition(int leftI, int rightI, int pivotI);
    void swap(int aI, int bI);
};

//KSE definitions:

KSE::KSE(vector<int> S, int K) {
    set = S;
    k = K;
    kthSmallest = findKthSmallest();
}

int KSE::findKthSmallest() {
    int leftI = 0;
    int rightI = set.size() - 1;
    int pivotI = -1;
    while (true) {
        if (leftI == rightI)
            break;//-----------------------------break

        //pivotI = rand() % (rightI - leftI + 1) + leftI - 1;//random number in the range of leftI to rightI, inclusive
        pivotI = leftI;
        cout << "pivot is " << set[pivotI] << endl;
        pivotI = partition(leftI, rightI, pivotI);

        cout << "set after partition: " << vecToStr(set) << endl;

        if (k == pivotI)
            break;
        else if (k < pivotI)
            rightI = pivotI - 1;
        else//if k > pivotI
            leftI = pivotI + 1;

        cout << "loopdone" << endl;
    }
    return set[k];
}

int KSE::partition(int leftI, int rightI, int pivotI) {
    //https://youtu.be/MZaf_9IZCrc
    int i = leftI - 1;
    for (int j = leftI; j <= rightI - 1; j++) {
        if (set[j] < set[pivotI]) {
            i++;
            swap(i, j);
        }
    }
    swap(i + 1, pivotI);
    pivotI = i + 1; 
    return pivotI;
}

void KSE::swap(int aI, int bI) {
    cout << "swapping " << set[aI] << " with " << set[bI] << endl;
    int tempV = set[aI];
    set[aI] = set[bI];
    set[bI] = tempV;
}

int main() {

    vector<int> kse_case1_set = { 2,3,5,8,12,1,7,10,13,30,6,14,15,18,22 };
    int kse_case1_k1 = 4;
    cout << "Testing Find Kth Element Algorithm" << endl << endl;
    cout << "set = " << vecToStr(kse_case1_set) << endl;

    KSE test1 = KSE(kse_case1_set, kse_case1_k1);
    cout << "when k = " << test1.k << endl;
    cout << "\tkth smallest element = " << test1.kthSmallest  << endl << endl;

    cout << "Press Enter to exit.";
    cin.get();
}

//to help with printing test cases:
string vecToStr(vector<int> vec) {
    string str = "[";
    for (int i{ 0 }; i < vec.size() - 1; i++) {
        str += to_string(vec[i]);
        str += ", ";
    }
    str += to_string(vec[vec.size() - 1]);
    str += "]";
    return str;
}

And here is what it outputs:

set = [2, 3, 5, 8, 12, 1, 7, 10, 13, 30, 6, 14, 15, 18, 22]
pivot is 2
swapping 2 with 1
swapping 3 with 1
set after partition: [3, 1, 5, 8, 12, 2, 7, 10, 13, 30, 6, 14, 15, 18, 22]
loopdone
pivot is 5
swapping 5 with 2
swapping 8 with 2
set after partition: [3, 1, 8, 2, 12, 5, 7, 10, 13, 30, 6, 14, 15, 18, 22]
loopdone
pivot is 12
swapping 12 with 5
swapping 12 with 5
set after partition: [3, 1, 8, 2, 12, 5, 7, 10, 13, 30, 6, 14, 15, 18, 22]
loopdone
when k = 4
        kth smallest element = 12

Press Enter to exit.

The 4th smallest element (starting at 0th) should be 6, not 12. Do you guys know what's going wrong?