Ridge with GLMNET: discrepancies

`glmnet` ridge does not seem to always give the same results as others on simple datasets (even outside of Benchopt, that's why I present the R code):
To solve the ridge problem: $\min \frac{1}{2}||y-X\beta||_2^2 + \frac{\lambda}{2}||\beta||_2^2$ the penalty in `glmnet` is: $\lambda_R=\frac{\sigma(y)}{n}\lambda$ with $\sigma(y)$ the french standard deviation

```r
set.seed(123)
n <- 1000
p <- 100
X <- matrix(rnorm(n*p,0,1),n,p)
beta <- rnorm(p,0,1)
Y <- X%*%beta+rnorm(n,0,0.5)
sd_y <- sqrt(var(Y)*(n-1)/n)[1,1]
beta1 <- solve(t(X)%*%X+10*diag(p),t(X)%*%(Y))[,1]
fit_glmnet <- glmnet(X,Y, alpha=0.001, standardize = F, intercept = FALSE, thresh = 1e-20, lambda=sd_y*10/n)
beta2 <- as.vector(coef(fit_glmnet))[-1]
cbind(beta1[1:10], beta2[1:10]) # same beta
```

But on datasets like `bodyfat`:
```r
X <- bodyfat[, 4:17]
X <- sapply(X, as.numeric)
Y <- bodyfat$siri
p <- dim(X)[2]
n <- dim(X)[1]
sd_y <- sqrt(var(Y)*(n-1)/n)
beta1 <- solve(t(X)%*%X+10*diag(p),t(X)%*%(Y))
fit_glmnet <- glmnet(X,Y, alpha=0.001, standardize = F, lambda=sd_y*10/n,
intercept = F, thresh = 1e-20, maxit=10000000)
beta2 <- as.vector(coef(fit_glmnet))[-1]
cbind(beta1[1:10], beta2[1:10]) # not exactly same beta
```

@cassiofragadantas found [this post](https://stats.stackexchange.com/questions/74206/ridge-regression-results-different-in-using-lm-ridge-and-glmnet) that also says there are discrepancies not explained by rescaling
So how would we treat this in Benchopt ? (a case where the solver does not converge to the same coefficients as others sometimes)