Revolutions increase and decrease

[AbdulBMWi3]

Engine revolutions fluctuate at 40 mph, however idle performance remains stable.

[DivineStripe]

Could it be the fuel pump?

[AbdulBMWi3]

Uncertain, since it operates well at lower speeds and when reaching 40, but maintaining that speed results in a sensation of stuttering; yet, applying more pressure on the accelerator rectifies the issue.

[dontito]

Initially, I would inspect for vacuum leaks, since they often do not trigger a check engine light. The automobiles do not seem to be "hunting" if this behaviour occurs alone at 40 mph.

Is it an auto? What is the engine size of the model make? Does it experience a decline in performance from a stationary position? Does softly touching the throttle at idle in neutral cause the rpm to decrease?

[AbdulBMWi3]

Indeed, it is a 2008 5 Series 2-liter diesel. It does not experience any bogging down during initial acceleration; however, at approximately 40 mph, the revs fluctuate between 2000 and 2100 RPM for an extended duration. While the variation is minimal, it is perceptible. Pressing the accelerator resolves the issue, but it has become increasingly bothersome.
Thank you for your assistance.

[DivineStripe]

What is the condition of the plugs? It may be beneficial to add some injectable cleanser to assess its efficacy.

[AbdulBMWi3]

I will attempt that, hoping it will be straightforward. Thank you so much.

[sherlockHOME]

What is the mileage? It may be advisable to replace the gearbox oil if the mileage exceeds around 70,000 miles. Attempt to engage manual mode on the vehicle and operate at 40 mph throughout several gears to see whether the issue continues.

[AbdulBMWi3]

The mileage is 85,000, hence I will include it in the list of issues to address, since it is likely overdue for a change. I will attempt to engage manual mode to observe the outcome.
Thank you for the counsel.

[AbdulBMWi3]