Is spurious regression a problem for lasso and similar techniques?

I was toying with R to see how the number of variables might affect spurious regression. Suppose that we have an $I(1)$ vector $y$ and a matrix $X$ with $I(1)$ columns. If the two are not related then OLS regression will be disastrous, with up to 50% of $X$‘S columns showing significance. On the other hand suppose I set

$$y =X_1beta + epsilon $$

where $X_1$ is the first column of the $X$ matrix and $epsilon$ is white noise. Then the regression works beautifully – the $y$ and $X_1$ form a cointegrating pair and the regression rightfully determines that the other columns are unrelated to the outcome, despite being nonstationary.

This begs the question – in situations where you have thousands or more variables and you would use regularized regression techniques, is spurious regression a problem? It seems that as long as there’s at least one variable related to the outcome your regression will be fine.

The code for my experiment:

    nruns <- 1000
    nobs <- 1000
    nvars <- 100
    significant_coefs <- numeric(nruns)
    
    for(i in 1:nruns) {
      X <- replicate(nvars, cumsum(rnorm(nobs)))
      y <- X[, 1] + rnorm(nobs, sd = 1000)
      
      model <- lm(y ~ X)
      significant_coefs[i] <- sum(summary(model)$coefficients[, 4] <=
                                    0.05)
    }
    
    hist(significant_coefs)

To see the impact of spurious regression just change the $y$ variable to a random walk.

    nruns <- 1000
    nobs <- 1000
    nvars <- 100
    significant_coefs <- numeric(nruns)
    
    for(i in 1:nruns) {
      X <- replicate(nvars, cumsum(rnorm(nobs)))
      y <- cumsum(rnorm(nobs))
      
      model <- lm(y ~ X)
      significant_coefs[i] <- sum(summary(model)$coefficients[, 4] <= 
                               0.05)
    }
    
    hist(significant_coefs)

In the first case I get an average of 6 coefficients with p-values less than 0.05, in the second I get 51.