Is this a good follow-up?

I recently applied to a job and I want to follow up because they haven't reached out about an interview yet. I have a potential follow up email I was hoping people here could let me know if it's good or bad. Before that, I want to give a brief summary of what has already transpired

January 3: I applied and asked my professor to let the hiring people know I applied since that's who I got the job posting from

January 9: They sent out an email with information for an online test we had to take for the job.

January 15: I sent a quick email saying I completed it and asking about next steps.

January 23: I got an email back saying they will touch base in a week or so about those chosen for the next round.

January 29: Got an email saying I passed the test, and that "We are now reviewing your resume and will let you know very soon if we will move you to a phone/skype interview.
Thank you for your continued interest and we will be in touch."

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I haven't heard anything and I know they said they will be in touch, but it's been a while and this is a job that I really want. I was going to send this as a response:

"Thank you so much for keeping me updated and letting me know I passed the initial test. I'm interviewing with a handful of organizations who have already expressed interested and [employer name] is one of my top choices so I hope we have the chance to connect directly soon."

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Does it sound too pushy? Should I not respond at all?

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