Confidence interval for standard deviation implemented but unsure if correct.

Question

tl;dr; why is does my monte carlo simulations differ so much with the confidence interval?

So I was checking the confidence interval for the sample standard deviation for a normal distribution: https://en.wikipedia.org/wiki/Standard_deviation#Confidence_interval_of_a_sampled_standard_deviation

Now I created a plot using Matlab where the confidence interval is displayed as function of the sample size. At first I thought I implemented it correctly.

Then, I also added some Monte Carlo simulations and plotted them. However, that kind of makes we wonder if my confidence interval is correctly implemented.

What goes wrong here?

The figure;

My Matlab code;

clear all;
close all;
clc;

% Number of samples
N = 2:50;

% Mean and standard deviation
mu    = 0;
sigma = 0.2;

% Confidence level
alpha = 0.975;

% Generate confidence interval
for i = 1:length(N)
    % Degrees of freedom
    k = N(i) - 1;
    
    % Calculate confidence interval
    lower(i) = k*sigma^2 / chi2inv(1 - alpha,k);
    upper(i) = k*sigma^2 / chi2inv(    alpha,k);
end

% Create figure
figure;
hold all;

% Plot confidence interval (standard deviation, not variance)
plot(N,sqrt(lower));
plot(N,sqrt(upper));

% Plot standard deviation
plot([N(1),N(end)],[sigma,sigma]);

% Monte carlo simulation
NN = []; 
XX = [];
% For each sample size
for i = 1:length(N)
    
    % Do 500 simulations
    for j = 1:500
        NN = [NN N(i)];
        XX = [XX std(normrnd(mu,sigma,1,N(i)))];
    end
end

scatter(NN,XX,15,'filled');
grid on; box on;
xlabel('Sample size N');
ylabel('Standard deviation [-]');
legend({'Upper confidence bound','Lower confidence bound','Standard deviation','Monte carlo simulation'});
ylim([0 0.6]);