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Let <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> denote an algebraically closed field; let <i>q</i> be a nonzero scalar in <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> such that <i>q</i> is not a root of unity; let <i>d</i> be a nonnegative integer; and let <i>X</i>, <i>Y</i>, <i>Z</i> be the equitable generators of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> over <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. Let <i>V</i> denote a finite-dimensional irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-module with dimension <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and let <i>R</i> denote the set of all linear maps from <i>V</i> to itself that act tridiagonally on the standard ordering of the eigenbases for each of <i>X</i>, <i>Y</i>, and <i>Z</i>. We show that <i>R</i> has dimension at most seven. Indeed, we show that the actions of 1, <i>X</i>, <i>Y</i>, <i>Z</i>, <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mi>Y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>Y</mi><mi>Z</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>Z</mi><mi>X</mi></mrow></semantics></math></inline-formula> on <i>V</i> give a basis for <i>R</i> when <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>.