Please help me solve (or disprove) this challenge! This is making me lose sleep at night and its so dumb.

At work I have some random legos that I fidget with and sometimes invent random challenges for my coworkers. The most recent completely made up challenge has driven me nuts and I can't stop thinking about it. Can someone either solve or confirm its unsolvable??

The challenge:

Given the following bricks can you create a completely connected structure with no red faces showing ie, the red bricks are completely covered (little slivers are fine...).

2 of 2x4 red bricks

4 of 2x4 blue bricks

2 of 2x3 blue bricks

2 of 2x2 blue bricks

3 of 1x2 blue bricks

You have to be able to pick up the connected structure and turn it around. From any side you look, you can't see any red faces--only blue!

If anyone figures this out or proves it is impossible I will be extremely grateful. We have gotten soo close, so many times!