I believe you can visit your page, set the local storage value, and then validate the checkbox. Roughly, it would look like:

cy.visit('/projects');
cy.window().then((window) => {
  window.localStorage.setItem('myLocalStorageItem', value);
})
cy.VerifyCheckBox('A', 'checked');

It doesn't mimic an exact user's workflow, but it works the same overall -- the checkbox is checked when a localStorage item is set to some value.

I would follow the pattern that @agoff uses, but use window:before:load event.

That way will catch any regression that clears the value during page load.

// verify that CheckboxA sets local storage
cy.visit('/projects')
cy.checkCheckbox('A')
cy.window().then((window) => {
  expect(window.localStorage.getItem('myLocalStorageItem')).to.eq(value)
})

// verify that reload retains localStorage value
cy.on('window:before:load', (win) => {      
  // should fire after Cypress clears localStorage
  window.localStorage.setItem('myLocalStorageItem', value)  
}) 
cy.reload()
cy.VerifyCheckBox('A', 'checked')

I have found a second solution for my problem. I added at the beginning of my test case (in describe, bevor it) this:

beforeEach(() => {
        cy.restoreLocalStorage().then(() => {
            cy.log('Storage restored');
        })
    });

    afterEach(() => {
        cy.saveLocalStorage().then(() => {
            cy.log('Storage saved');
        })
    });

I hope I can help you with it! Let me know it!