Better way to update an object's value at a variable depth

I am working on some software that reads/writes information in localStorage using a handler. You can find a working example here: http://jsbin.com/wifucugoko/edit?js,console

My problem is with the segment of code below (focusing on the switch statement):

_t.set = function(path, value) { // Update a single value or object
        if (~path.indexOf(".")) {
            let o = path.split(".")[0],
                p = this.get(o),
                q = path.split(".").slice(1);
            switch (q.length) { 
            // There has to be a better way to do this... 
                case 1:
                    p[q[0]] = value; 
                    break;
                case 2:
                    p[q[0]][q[1]] = value; 
                    break;
                case 3:
                    p[q[0]][q[1]][q[2]] = value; 
                    break;
                case 4:
                    p[q[0]][q[1]][q[2]][q[3]] = value; 
                    break;
                case 5:
                    p[q[0]][q[1]][q[2]][q[3]][q[4]] = value; 
                    break;
                case 6: 
                    p[q[0]][q[1]][q[2]][q[3]][q[4]][q[5]] = value; 
                    break;
                default:  
                    return "error";
            }  
            b.setItem(o, JSON.stringify(p));
            return p;
        } else {
           b.setItem(path, JSON.stringify(value));
            return this.get(path);
        }
    };

I am not going to be the only one using this codebase, and I am trying to make it easy for others to update any value that could be placed in localStorage. Right now you can update a value by using something like local.set('item.subitem.proeprty', 'value') Though the code above does that, it's ugly and doesn't scale.

How can this method be improved to (1) update a property nested at any depth automatically, instead of writing an infinitely-long switch statement, and (2) not lace a parent object with [object Object] after a value is updated?

---

This question has nothing to do with my use of localStorage. I originally posted this question in code review, which requires a working contextual example. They closed this question immediately, since part of my problem is the code I provided doesn't work once you start dealing with updating a value nested more than six objects deep. Though I could have continued my switch statement indefinitely, that's exactly what I'm trying to avoid.

With the three examples provided you'll see that setting a value in one place doesn't remove values in other places:

 local.set('user.session.timeout', false);
 local.set('user.name', {first:'john', last:'doe', mi:'c'});
 local.set('user.PIN', 8675309);

All these values, though set at different times, only UPDATE or create a value, they do NOT clear any pre-existing values elsewhere.

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