Defect relation for points in $\mathbb P^2$

Definitions and mathematical statement

Let $f:\mathbb C\to\mathbb P^2(\mathbb C)$ be a holomorphic curve with an entire homogeneous lift $F=(F_0,F_1,F_2)$ having no common zero. Require linear nondegeneracy: the functions $F_j$ are linearly independent over $\mathbb C$, equivalently the image lies in no projective line. Using the Euclidean norm on $\mathbb C^3$, define \[ T(r,f)=\frac1{2\pi}\int_0^{2\pi} \log\|F(re^{it})\|\,dt-\log\|F(0)\|. \] This characteristic is independent of the chosen zero-free scalar change of lift.

For a point $a=[A]\in\mathbb P^2$, set $d(f(z),a)=\|F(z)\wedge A\|/(\|F(z)\|\|A\|)$, the sine of the Fubini--Study distance. Define \[ m(r,a,f)=\frac1{2\pi}\int_0^{2\pi} \log\frac1{d(f(re^{it}),a)}\,dt, \qquad \delta(a,f)=\liminf_{r\to\infty}\frac{m(r,a,f)}{T(r,f)}. \] The logarithmic singularities at isolated preimages are interpreted in the integral sense. Replacing the sine distance by the ordinary Fubini--Study distance changes the proximity by a bounded term, and leaves this deficiency unchanged.

For every finite set of distinct points in general position, meaning that no three are collinear, must $\sum_a\delta(a,f)\leq1$? The same formulation for an infinite family uses the supremum of its finite subsums. The targets are points, of codimension two, rather than hyperplanes. The question is the sharp simultaneous defect bound for every linearly nondegenerate curve.

Short English statement

Is the total deficiency of points in general position at most one for every linearly nondegenerate entire curve in the projective plane?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700040 (AMR-036-0040), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Defect relation conjecture. Original source indexed by AMR-036-0040. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Defect relation for targets of large codimension. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/points.pdf.